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Commutators of small rank and reducibility of operator semigroups


Authors: Ali Jafarian, Alexey I. Popov, Mehdi Radjabalipour and Heydar Radjavi
Journal: Proc. Amer. Math. Soc. 142 (2014), 4277-4289
MSC (2010): Primary 47D03, 20M20; Secondary 47B47, 51F25
DOI: https://doi.org/10.1090/S0002-9939-2014-12217-9
Published electronically: August 13, 2014
MathSciNet review: 3266995
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Abstract: It is easy to see that if $ \mathcal {G}$ is a non-abelian group of unitary matrices, then for no members $ A$ and $ B$ of $ \mathcal {G}$ can the rank of $ AB-BA$ be one. We examine the consequences of the assumption that this rank is at most two for a general semigroup $ \mathcal {S}$ of linear operators. Our conclusion is that under obviously necessary, but trivial, size conditions, $ \mathcal {S}$ is reducible. In the case of a unitary group satisfying the hypothesis, we show that it is contained in the direct sum $ \mathcal {G}_1\oplus \mathcal {G}_2$, where $ \mathcal {G}_1$ is at most $ 3\times 3$ and $ \mathcal {G}_2$ is abelian.


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Additional Information

Ali Jafarian
Affiliation: University of New Haven, 300 Boston Post Road, West Haven, Connecticut 06516
Email: ajafarian@newhaven.edu

Alexey I. Popov
Affiliation: Department of Pure Mathematics, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, N2L3G1, Canada
Email: a4popov@uwaterloo.ca

Mehdi Radjabalipour
Affiliation: Department of Pure Mathematics, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, N2L3G1, Canada (on sabbatical from the Iranian Academy of Sciences, Tehran, Iran)
Email: radjabalipour@ias.ac.ir

Heydar Radjavi
Affiliation: Department of Pure Mathematics, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, N2L3G1, Canada
Email: hradjavi@uwaterloo.ca

DOI: https://doi.org/10.1090/S0002-9939-2014-12217-9
Keywords: Semigroup of operators, unitary group, commutator, rank, invariant subspace
Received by editor(s): January 16, 2013
Published electronically: August 13, 2014
Additional Notes: The second and fourth authors’ research was supported in part by NSERC (Canada)
The third author’s research was supported in part by the Iranian National Science Foundation
Communicated by: Pamela B. Gorkin
Article copyright: © Copyright 2014 American Mathematical Society